src/HOL/Library/Binomial.thy
author Christian Sternagel
Wed Aug 29 12:23:14 2012 +0900 (2012-08-29)
changeset 49083 01081bca31b6
parent 48830 72efe3e0a46b
child 50224 aacd6da09825
permissions -rw-r--r--
dropped ord and bot instance for list prefixes (use locale interpretation instead, which allows users to decide what order to use on lists)
     1 (*  Title:      HOL/Library/Binomial.thy
     2     Author:     Lawrence C Paulson, Amine Chaieb
     3     Copyright   1997  University of Cambridge
     4 *)
     5 
     6 header {* Binomial Coefficients *}
     7 
     8 theory Binomial
     9 imports Complex_Main
    10 begin
    11 
    12 text {* This development is based on the work of Andy Gordon and
    13   Florian Kammueller. *}
    14 
    15 primrec binomial :: "nat \<Rightarrow> nat \<Rightarrow> nat" (infixl "choose" 65) where
    16   binomial_0: "(0 choose k) = (if k = 0 then 1 else 0)"
    17 | binomial_Suc: "(Suc n choose k) =
    18                  (if k = 0 then 1 else (n choose (k - 1)) + (n choose k))"
    19 
    20 lemma binomial_n_0 [simp]: "(n choose 0) = 1"
    21   by (cases n) simp_all
    22 
    23 lemma binomial_0_Suc [simp]: "(0 choose Suc k) = 0"
    24   by simp
    25 
    26 lemma binomial_Suc_Suc [simp]:
    27   "(Suc n choose Suc k) = (n choose k) + (n choose Suc k)"
    28   by simp
    29 
    30 lemma binomial_eq_0: "!!k. n < k ==> (n choose k) = 0"
    31   by (induct n) auto
    32 
    33 declare binomial_0 [simp del] binomial_Suc [simp del]
    34 
    35 lemma binomial_n_n [simp]: "(n choose n) = 1"
    36   by (induct n) (simp_all add: binomial_eq_0)
    37 
    38 lemma binomial_Suc_n [simp]: "(Suc n choose n) = Suc n"
    39   by (induct n) simp_all
    40 
    41 lemma binomial_1 [simp]: "(n choose Suc 0) = n"
    42   by (induct n) simp_all
    43 
    44 lemma zero_less_binomial: "k \<le> n ==> (n choose k) > 0"
    45   by (induct n k rule: diff_induct) simp_all
    46 
    47 lemma binomial_eq_0_iff: "(n choose k = 0) = (n<k)"
    48   apply (safe intro!: binomial_eq_0)
    49   apply (erule contrapos_pp)
    50   apply (simp add: zero_less_binomial)
    51   done
    52 
    53 lemma zero_less_binomial_iff: "(n choose k > 0) = (k\<le>n)"
    54   by (simp add: linorder_not_less binomial_eq_0_iff neq0_conv[symmetric] del: neq0_conv)
    55 
    56 (*Might be more useful if re-oriented*)
    57 lemma Suc_times_binomial_eq:
    58   "!!k. k \<le> n ==> Suc n * (n choose k) = (Suc n choose Suc k) * Suc k"
    59   apply (induct n)
    60    apply (simp add: binomial_0)
    61    apply (case_tac k)
    62   apply (auto simp add: add_mult_distrib add_mult_distrib2 le_Suc_eq binomial_eq_0)
    63   done
    64 
    65 text{*This is the well-known version, but it's harder to use because of the
    66   need to reason about division.*}
    67 lemma binomial_Suc_Suc_eq_times:
    68     "k \<le> n ==> (Suc n choose Suc k) = (Suc n * (n choose k)) div Suc k"
    69   by (simp add: Suc_times_binomial_eq del: mult_Suc mult_Suc_right)
    70 
    71 text{*Another version, with -1 instead of Suc.*}
    72 lemma times_binomial_minus1_eq:
    73     "[|k \<le> n;  0<k|] ==> (n choose k) * k = n * ((n - 1) choose (k - 1))"
    74   apply (cut_tac n = "n - 1" and k = "k - 1" in Suc_times_binomial_eq)
    75    apply (simp split add: nat_diff_split, auto)
    76   done
    77 
    78 
    79 subsection {* Theorems about @{text "choose"} *}
    80 
    81 text {*
    82   \medskip Basic theorem about @{text "choose"}.  By Florian
    83   Kamm\"uller, tidied by LCP.
    84 *}
    85 
    86 lemma card_s_0_eq_empty: "finite A ==> card {B. B \<subseteq> A & card B = 0} = 1"
    87   by (simp cong add: conj_cong add: finite_subset [THEN card_0_eq])
    88 
    89 lemma choose_deconstruct: "finite M ==> x \<notin> M
    90   ==> {s. s <= insert x M & card(s) = Suc k}
    91        = {s. s <= M & card(s) = Suc k} Un
    92          {s. EX t. t <= M & card(t) = k & s = insert x t}"
    93   apply safe
    94      apply (auto intro: finite_subset [THEN card_insert_disjoint])
    95   apply (drule_tac x = "xa - {x}" in spec)
    96   apply (subgoal_tac "x \<notin> xa", auto)
    97   apply (erule rev_mp, subst card_Diff_singleton)
    98     apply (auto intro: finite_subset)
    99   done
   100 (*
   101 lemma "finite(UN y. {x. P x y})"
   102 apply simp
   103 lemma Collect_ex_eq
   104 
   105 lemma "{x. EX y. P x y} = (UN y. {x. P x y})"
   106 apply blast
   107 *)
   108 
   109 lemma finite_bex_subset[simp]:
   110   "finite B \<Longrightarrow> (!!A. A<=B \<Longrightarrow> finite{x. P x A}) \<Longrightarrow> finite{x. EX A<=B. P x A}"
   111   apply (subgoal_tac "{x. EX A<=B. P x A} = (UN A:Pow B. {x. P x A})")
   112    apply simp
   113   apply blast
   114   done
   115 
   116 text{*There are as many subsets of @{term A} having cardinality @{term k}
   117  as there are sets obtained from the former by inserting a fixed element
   118  @{term x} into each.*}
   119 lemma constr_bij:
   120    "[|finite A; x \<notin> A|] ==>
   121     card {B. EX C. C <= A & card(C) = k & B = insert x C} =
   122     card {B. B <= A & card(B) = k}"
   123   apply (rule_tac f = "%s. s - {x}" and g = "insert x" in card_bij_eq)
   124        apply (auto elim!: equalityE simp add: inj_on_def)
   125   apply (subst Diff_insert0, auto)
   126   done
   127 
   128 text {*
   129   Main theorem: combinatorial statement about number of subsets of a set.
   130 *}
   131 
   132 lemma n_sub_lemma:
   133     "!!A. finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
   134   apply (induct k)
   135    apply (simp add: card_s_0_eq_empty, atomize)
   136   apply (rotate_tac -1, erule finite_induct)
   137    apply (simp_all (no_asm_simp) cong add: conj_cong
   138      add: card_s_0_eq_empty choose_deconstruct)
   139   apply (subst card_Un_disjoint)
   140      prefer 4 apply (force simp add: constr_bij)
   141     prefer 3 apply force
   142    prefer 2 apply (blast intro: finite_Pow_iff [THEN iffD2]
   143      finite_subset [of _ "Pow (insert x F)", standard])
   144   apply (blast intro: finite_Pow_iff [THEN iffD2, THEN [2] finite_subset])
   145   done
   146 
   147 theorem n_subsets:
   148     "finite A ==> card {B. B <= A & card B = k} = (card A choose k)"
   149   by (simp add: n_sub_lemma)
   150 
   151 
   152 text{* The binomial theorem (courtesy of Tobias Nipkow): *}
   153 
   154 theorem binomial: "(a+b::nat)^n = (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
   155 proof (induct n)
   156   case 0 thus ?case by simp
   157 next
   158   case (Suc n)
   159   have decomp: "{0..n+1} = {0} \<union> {n+1} \<union> {1..n}"
   160     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
   161   have decomp2: "{0..n} = {0} \<union> {1..n}"
   162     by (auto simp add:atLeastAtMost_def atLeast_def atMost_def)
   163   have "(a+b::nat)^(n+1) = (a+b) * (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
   164     using Suc by simp
   165   also have "\<dots> =  a*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k)) +
   166                    b*(\<Sum>k=0..n. (n choose k) * a^k * b^(n-k))"
   167     by (rule nat_distrib)
   168   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^(k+1) * b^(n-k)) +
   169                   (\<Sum>k=0..n. (n choose k) * a^k * b^(n-k+1))"
   170     by (simp add: setsum_right_distrib mult_ac)
   171   also have "\<dots> = (\<Sum>k=0..n. (n choose k) * a^k * b^(n+1-k)) +
   172                   (\<Sum>k=1..n+1. (n choose (k - 1)) * a^k * b^(n+1-k))"
   173     by (simp add:setsum_shift_bounds_cl_Suc_ivl Suc_diff_le
   174              del:setsum_cl_ivl_Suc)
   175   also have "\<dots> = a^(n+1) + b^(n+1) +
   176                   (\<Sum>k=1..n. (n choose (k - 1)) * a^k * b^(n+1-k)) +
   177                   (\<Sum>k=1..n. (n choose k) * a^k * b^(n+1-k))"
   178     by (simp add: decomp2)
   179   also have
   180       "\<dots> = a^(n+1) + b^(n+1) + (\<Sum>k=1..n. (n+1 choose k) * a^k * b^(n+1-k))"
   181     by (simp add: nat_distrib setsum_addf binomial.simps)
   182   also have "\<dots> = (\<Sum>k=0..n+1. (n+1 choose k) * a^k * b^(n+1-k))"
   183     using decomp by simp
   184   finally show ?case by simp
   185 qed
   186 
   187 subsection{* Pochhammer's symbol : generalized raising factorial*}
   188 
   189 definition "pochhammer (a::'a::comm_semiring_1) n = (if n = 0 then 1 else setprod (\<lambda>n. a + of_nat n) {0 .. n - 1})"
   190 
   191 lemma pochhammer_0[simp]: "pochhammer a 0 = 1"
   192   by (simp add: pochhammer_def)
   193 
   194 lemma pochhammer_1[simp]: "pochhammer a 1 = a" by (simp add: pochhammer_def)
   195 lemma pochhammer_Suc0[simp]: "pochhammer a (Suc 0) = a"
   196   by (simp add: pochhammer_def)
   197 
   198 lemma pochhammer_Suc_setprod: "pochhammer a (Suc n) = setprod (\<lambda>n. a + of_nat n) {0 .. n}"
   199   by (simp add: pochhammer_def)
   200 
   201 lemma setprod_nat_ivl_Suc: "setprod f {0 .. Suc n} = setprod f {0..n} * f (Suc n)"
   202 proof-
   203   have eq: "{0..Suc n} = {0..n} \<union> {Suc n}" by auto
   204   show ?thesis unfolding eq by (simp add: field_simps)
   205 qed
   206 
   207 lemma setprod_nat_ivl_1_Suc: "setprod f {0 .. Suc n} = f 0 * setprod f {1.. Suc n}"
   208 proof-
   209   have eq: "{0..Suc n} = {0} \<union> {1 .. Suc n}" by auto
   210   show ?thesis unfolding eq by simp
   211 qed
   212 
   213 
   214 lemma pochhammer_Suc: "pochhammer a (Suc n) = pochhammer a n * (a + of_nat n)"
   215 proof-
   216   { assume "n=0" then have ?thesis by simp }
   217   moreover
   218   { fix m assume m: "n = Suc m"
   219     have ?thesis unfolding m pochhammer_Suc_setprod setprod_nat_ivl_Suc .. }
   220   ultimately show ?thesis by (cases n) auto
   221 qed
   222 
   223 lemma pochhammer_rec: "pochhammer a (Suc n) = a * pochhammer (a + 1) n"
   224 proof-
   225   { assume "n=0" then have ?thesis by (simp add: pochhammer_Suc_setprod) }
   226   moreover
   227   { assume n0: "n \<noteq> 0"
   228     have th0: "finite {1 .. n}" "0 \<notin> {1 .. n}" by auto
   229     have eq: "insert 0 {1 .. n} = {0..n}" by auto
   230     have th1: "(\<Prod>n\<in>{1\<Colon>nat..n}. a + of_nat n) =
   231       (\<Prod>n\<in>{0\<Colon>nat..n - 1}. a + 1 + of_nat n)"
   232       apply (rule setprod_reindex_cong [where f = Suc])
   233       using n0 by (auto simp add: fun_eq_iff field_simps)
   234     have ?thesis apply (simp add: pochhammer_def)
   235     unfolding setprod_insert[OF th0, unfolded eq]
   236     using th1 by (simp add: field_simps) }
   237   ultimately show ?thesis by blast
   238 qed
   239 
   240 lemma pochhammer_fact: "of_nat (fact n) = pochhammer 1 n"
   241   unfolding fact_altdef_nat
   242   apply (cases n)
   243    apply (simp_all add: of_nat_setprod pochhammer_Suc_setprod)
   244   apply (rule setprod_reindex_cong[where f=Suc])
   245     apply (auto simp add: fun_eq_iff)
   246   done
   247 
   248 lemma pochhammer_of_nat_eq_0_lemma:
   249   assumes kn: "k > n"
   250   shows "pochhammer (- (of_nat n :: 'a:: idom)) k = 0"
   251 proof-
   252   from kn obtain h where h: "k = Suc h" by (cases k) auto
   253   { assume n0: "n=0" then have ?thesis using kn
   254       by (cases k) (simp_all add: pochhammer_rec) }
   255   moreover
   256   { assume n0: "n \<noteq> 0"
   257     then have ?thesis
   258       apply (simp add: h pochhammer_Suc_setprod)
   259       apply (rule_tac x="n" in bexI)
   260       using h kn
   261       apply auto
   262       done }
   263   ultimately show ?thesis by blast
   264 qed
   265 
   266 lemma pochhammer_of_nat_eq_0_lemma': assumes kn: "k \<le> n"
   267   shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k \<noteq> 0"
   268 proof-
   269   { assume "k=0" then have ?thesis by simp }
   270   moreover
   271   { fix h assume h: "k = Suc h"
   272     then have ?thesis apply (simp add: pochhammer_Suc_setprod)
   273       using h kn by (auto simp add: algebra_simps) }
   274   ultimately show ?thesis by (cases k) auto
   275 qed
   276 
   277 lemma pochhammer_of_nat_eq_0_iff:
   278   shows "pochhammer (- (of_nat n :: 'a:: {idom, ring_char_0})) k = 0 \<longleftrightarrow> k > n"
   279   (is "?l = ?r")
   280   using pochhammer_of_nat_eq_0_lemma[of n k, where ?'a='a]
   281     pochhammer_of_nat_eq_0_lemma'[of k n, where ?'a = 'a]
   282   by (auto simp add: not_le[symmetric])
   283 
   284 
   285 lemma pochhammer_eq_0_iff:
   286   "pochhammer a n = (0::'a::field_char_0) \<longleftrightarrow> (EX k < n . a = - of_nat k) "
   287   apply (auto simp add: pochhammer_of_nat_eq_0_iff)
   288   apply (cases n)
   289    apply (auto simp add: pochhammer_def algebra_simps group_add_class.eq_neg_iff_add_eq_0)
   290   apply (rule_tac x=x in exI)
   291   apply auto
   292   done
   293 
   294 
   295 lemma pochhammer_eq_0_mono:
   296   "pochhammer a n = (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a m = 0"
   297   unfolding pochhammer_eq_0_iff by auto
   298 
   299 lemma pochhammer_neq_0_mono:
   300   "pochhammer a m \<noteq> (0::'a::field_char_0) \<Longrightarrow> m \<ge> n \<Longrightarrow> pochhammer a n \<noteq> 0"
   301   unfolding pochhammer_eq_0_iff by auto
   302 
   303 lemma pochhammer_minus:
   304   assumes kn: "k \<le> n"
   305   shows "pochhammer (- b) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (b - of_nat k + 1) k"
   306 proof-
   307   { assume k0: "k = 0" then have ?thesis by simp }
   308   moreover
   309   { fix h assume h: "k = Suc h"
   310     have eq: "((- 1) ^ Suc h :: 'a) = setprod (%i. - 1) {0 .. h}"
   311       using setprod_constant[where A="{0 .. h}" and y="- 1 :: 'a"]
   312       by auto
   313     have ?thesis
   314       unfolding h pochhammer_Suc_setprod eq setprod_timesf[symmetric]
   315       apply (rule strong_setprod_reindex_cong[where f = "%i. h - i"])
   316       apply (auto simp add: inj_on_def image_def h )
   317       apply (rule_tac x="h - x" in bexI)
   318       apply (auto simp add: fun_eq_iff h of_nat_diff)
   319       done }
   320   ultimately show ?thesis by (cases k) auto
   321 qed
   322 
   323 lemma pochhammer_minus':
   324   assumes kn: "k \<le> n"
   325   shows "pochhammer (b - of_nat k + 1) k = ((- 1) ^ k :: 'a::comm_ring_1) * pochhammer (- b) k"
   326   unfolding pochhammer_minus[OF kn, where b=b]
   327   unfolding mult_assoc[symmetric]
   328   unfolding power_add[symmetric]
   329   apply simp
   330   done
   331 
   332 lemma pochhammer_same: "pochhammer (- of_nat n) n = ((- 1) ^ n :: 'a::comm_ring_1) * of_nat (fact n)"
   333   unfolding pochhammer_minus[OF le_refl[of n]]
   334   by (simp add: of_nat_diff pochhammer_fact)
   335 
   336 subsection{* Generalized binomial coefficients *}
   337 
   338 definition gbinomial :: "'a::field_char_0 \<Rightarrow> nat \<Rightarrow> 'a" (infixl "gchoose" 65)
   339   where "a gchoose n =
   340     (if n = 0 then 1 else (setprod (\<lambda>i. a - of_nat i) {0 .. n - 1}) / of_nat (fact n))"
   341 
   342 lemma gbinomial_0[simp]: "a gchoose 0 = 1" "0 gchoose (Suc n) = 0"
   343   apply (simp_all add: gbinomial_def)
   344   apply (subgoal_tac "(\<Prod>i\<Colon>nat\<in>{0\<Colon>nat..n}. - of_nat i) = (0::'b)")
   345    apply (simp del:setprod_zero_iff)
   346   apply simp
   347   done
   348 
   349 lemma gbinomial_pochhammer: "a gchoose n = (- 1) ^ n * pochhammer (- a) n / of_nat (fact n)"
   350 proof -
   351   { assume "n=0" then have ?thesis by simp }
   352   moreover
   353   { assume n0: "n\<noteq>0"
   354     from n0 setprod_constant[of "{0 .. n - 1}" "- (1:: 'a)"]
   355     have eq: "(- (1\<Colon>'a)) ^ n = setprod (\<lambda>i. - 1) {0 .. n - 1}"
   356       by auto
   357     from n0 have ?thesis
   358       by (simp add: pochhammer_def gbinomial_def field_simps
   359         eq setprod_timesf[symmetric] del: minus_one) (* FIXME: del: minus_one *) }
   360   ultimately show ?thesis by blast
   361 qed
   362 
   363 lemma binomial_fact_lemma: "k \<le> n \<Longrightarrow> fact k * fact (n - k) * (n choose k) = fact n"
   364 proof (induct n arbitrary: k rule: nat_less_induct)
   365   fix n k assume H: "\<forall>m<n. \<forall>x\<le>m. fact x * fact (m - x) * (m choose x) =
   366                       fact m" and kn: "k \<le> n"
   367   let ?ths = "fact k * fact (n - k) * (n choose k) = fact n"
   368   { assume "n=0" then have ?ths using kn by simp }
   369   moreover
   370   { assume "k=0" then have ?ths using kn by simp }
   371   moreover
   372   { assume nk: "n=k" then have ?ths by simp }
   373   moreover
   374   { fix m h assume n: "n = Suc m" and h: "k = Suc h" and hm: "h < m"
   375     from n have mn: "m < n" by arith
   376     from hm have hm': "h \<le> m" by arith
   377     from hm h n kn have km: "k \<le> m" by arith
   378     have "m - h = Suc (m - Suc h)" using  h km hm by arith
   379     with km h have th0: "fact (m - h) = (m - h) * fact (m - k)"
   380       by simp
   381     from n h th0
   382     have "fact k * fact (n - k) * (n choose k) =
   383         k * (fact h * fact (m - h) * (m choose h)) +  (m - h) * (fact k * fact (m - k) * (m choose k))"
   384       by (simp add: field_simps)
   385     also have "\<dots> = (k + (m - h)) * fact m"
   386       using H[rule_format, OF mn hm'] H[rule_format, OF mn km]
   387       by (simp add: field_simps)
   388     finally have ?ths using h n km by simp }
   389   moreover have "n=0 \<or> k = 0 \<or> k = n \<or> (EX m h. n=Suc m \<and> k = Suc h \<and> h < m)"
   390     using kn by presburger
   391   ultimately show ?ths by blast
   392 qed
   393 
   394 lemma binomial_fact:
   395   assumes kn: "k \<le> n"
   396   shows "(of_nat (n choose k) :: 'a::field_char_0) =
   397     of_nat (fact n) / (of_nat (fact k) * of_nat (fact (n - k)))"
   398   using binomial_fact_lemma[OF kn]
   399   by (simp add: field_simps of_nat_mult [symmetric])
   400 
   401 lemma binomial_gbinomial: "of_nat (n choose k) = of_nat n gchoose k"
   402 proof -
   403   { assume kn: "k > n"
   404     from kn binomial_eq_0[OF kn] have ?thesis
   405       by (simp add: gbinomial_pochhammer field_simps  pochhammer_of_nat_eq_0_iff) }
   406   moreover
   407   { assume "k=0" then have ?thesis by simp }
   408   moreover
   409   { assume kn: "k \<le> n" and k0: "k\<noteq> 0"
   410     from k0 obtain h where h: "k = Suc h" by (cases k) auto
   411     from h
   412     have eq:"(- 1 :: 'a) ^ k = setprod (\<lambda>i. - 1) {0..h}"
   413       by (subst setprod_constant, auto)
   414     have eq': "(\<Prod>i\<in>{0..h}. of_nat n + - (of_nat i :: 'a)) = (\<Prod>i\<in>{n - h..n}. of_nat i)"
   415       apply (rule strong_setprod_reindex_cong[where f="op - n"])
   416         using h kn
   417         apply (simp_all add: inj_on_def image_iff Bex_def set_eq_iff)
   418         apply clarsimp
   419         apply presburger
   420        apply presburger
   421       apply (simp add: fun_eq_iff field_simps of_nat_add[symmetric] del: of_nat_add)
   422       done
   423     have th0: "finite {1..n - Suc h}" "finite {n - h .. n}"
   424         "{1..n - Suc h} \<inter> {n - h .. n} = {}" and
   425         eq3: "{1..n - Suc h} \<union> {n - h .. n} = {1..n}"
   426       using h kn by auto
   427     from eq[symmetric]
   428     have ?thesis using kn
   429       apply (simp add: binomial_fact[OF kn, where ?'a = 'a]
   430         gbinomial_pochhammer field_simps pochhammer_Suc_setprod del: minus_one)
   431       apply (simp add: pochhammer_Suc_setprod fact_altdef_nat h
   432         of_nat_setprod setprod_timesf[symmetric] eq' del: One_nat_def power_Suc del: minus_one)
   433       unfolding setprod_Un_disjoint[OF th0, unfolded eq3, of "of_nat:: nat \<Rightarrow> 'a"] eq[unfolded h]
   434       unfolding mult_assoc[symmetric]
   435       unfolding setprod_timesf[symmetric]
   436       apply simp
   437       apply (rule strong_setprod_reindex_cong[where f= "op - n"])
   438         apply (auto simp add: inj_on_def image_iff Bex_def)
   439        apply presburger
   440       apply (subgoal_tac "(of_nat (n - x) :: 'a) = of_nat n - of_nat x")
   441        apply simp
   442       apply (rule of_nat_diff)
   443       apply simp
   444       done
   445   }
   446   moreover
   447   have "k > n \<or> k = 0 \<or> (k \<le> n \<and> k \<noteq> 0)" by arith
   448   ultimately show ?thesis by blast
   449 qed
   450 
   451 lemma gbinomial_1[simp]: "a gchoose 1 = a"
   452   by (simp add: gbinomial_def)
   453 
   454 lemma gbinomial_Suc0[simp]: "a gchoose (Suc 0) = a"
   455   by (simp add: gbinomial_def)
   456 
   457 lemma gbinomial_mult_1:
   458   "a * (a gchoose n) =
   459     of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"  (is "?l = ?r")
   460 proof -
   461   have "?r = ((- 1) ^n * pochhammer (- a) n / of_nat (fact n)) * (of_nat n - (- a + of_nat n))"
   462     unfolding gbinomial_pochhammer
   463       pochhammer_Suc fact_Suc of_nat_mult right_diff_distrib power_Suc
   464     by (simp add:  field_simps del: of_nat_Suc)
   465   also have "\<dots> = ?l" unfolding gbinomial_pochhammer
   466     by (simp add: field_simps)
   467   finally show ?thesis ..
   468 qed
   469 
   470 lemma gbinomial_mult_1':
   471     "(a gchoose n) * a = of_nat n * (a gchoose n) + of_nat (Suc n) * (a gchoose (Suc n))"
   472   by (simp add: mult_commute gbinomial_mult_1)
   473 
   474 lemma gbinomial_Suc:
   475     "a gchoose (Suc k) = (setprod (\<lambda>i. a - of_nat i) {0 .. k}) / of_nat (fact (Suc k))"
   476   by (simp add: gbinomial_def)
   477 
   478 lemma gbinomial_mult_fact:
   479   "(of_nat (fact (Suc k)) :: 'a) * ((a::'a::field_char_0) gchoose (Suc k)) =
   480     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
   481   by (simp_all add: gbinomial_Suc field_simps del: fact_Suc)
   482 
   483 lemma gbinomial_mult_fact':
   484   "((a::'a::field_char_0) gchoose (Suc k)) * (of_nat (fact (Suc k)) :: 'a) =
   485     (setprod (\<lambda>i. a - of_nat i) {0 .. k})"
   486   using gbinomial_mult_fact[of k a]
   487   apply (subst mult_commute)
   488   apply assumption
   489   done
   490 
   491 
   492 lemma gbinomial_Suc_Suc:
   493   "((a::'a::field_char_0) + 1) gchoose (Suc k) = a gchoose k + (a gchoose (Suc k))"
   494 proof -
   495   { assume "k = 0" then have ?thesis by simp }
   496   moreover
   497   { fix h assume h: "k = Suc h"
   498     have eq0: "(\<Prod>i\<in>{1..k}. (a + 1) - of_nat i) = (\<Prod>i\<in>{0..h}. a - of_nat i)"
   499       apply (rule strong_setprod_reindex_cong[where f = Suc])
   500         using h
   501         apply auto
   502       done
   503 
   504     have "of_nat (fact (Suc k)) * (a gchoose k + (a gchoose (Suc k))) =
   505       ((a gchoose Suc h) * of_nat (fact (Suc h)) * of_nat (Suc k)) + (\<Prod>i\<in>{0\<Colon>nat..Suc h}. a - of_nat i)"
   506       apply (simp add: h field_simps del: fact_Suc)
   507       unfolding gbinomial_mult_fact'
   508       apply (subst fact_Suc)
   509       unfolding of_nat_mult
   510       apply (subst mult_commute)
   511       unfolding mult_assoc
   512       unfolding gbinomial_mult_fact
   513       apply (simp add: field_simps)
   514       done
   515     also have "\<dots> = (\<Prod>i\<in>{0..h}. a - of_nat i) * (a + 1)"
   516       unfolding gbinomial_mult_fact' setprod_nat_ivl_Suc
   517       by (simp add: field_simps h)
   518     also have "\<dots> = (\<Prod>i\<in>{0..k}. (a + 1) - of_nat i)"
   519       using eq0
   520       by (simp add: h setprod_nat_ivl_1_Suc)
   521     also have "\<dots> = of_nat (fact (Suc k)) * ((a + 1) gchoose (Suc k))"
   522       unfolding gbinomial_mult_fact ..
   523     finally have ?thesis by (simp del: fact_Suc)
   524   }
   525   ultimately show ?thesis by (cases k) auto
   526 qed
   527 
   528 
   529 lemma binomial_symmetric:
   530   assumes kn: "k \<le> n"
   531   shows "n choose k = n choose (n - k)"
   532 proof-
   533   from kn have kn': "n - k \<le> n" by arith
   534   from binomial_fact_lemma[OF kn] binomial_fact_lemma[OF kn']
   535   have "fact k * fact (n - k) * (n choose k) =
   536     fact (n - k) * fact (n - (n - k)) * (n choose (n - k))" by simp
   537   then show ?thesis using kn by simp
   538 qed
   539 
   540 end